派生証券の価格計算

最近金融の計算方式を味わっています。数学が特に役に立つところが派生証券のリスク計算だと思います。
Matlab is used to automate computations, remember the results, and visualize output. It provides an interpreter and scripting front end to what earlier scientific applications might have implemented via custom Fortran modules.
Differential equations are written in terms of derivatives, which represent the rate of change of one variable with respect to a unit change in another variable. Analytic solutions to differential equations are often obtained via integration. Both derivatives and integrals can be computed analytically in Matlab, provided that the function to be operated on is a polynomial of one variable.
Two differential equations utilized in the pricing of derivatives are the Black-Scholes differential equation and the Hull-White formula.
The Black-Scholes equation is:
Cr = theta + rS(delta) + (1/2)(sigma**2)(S**2)gamma
C is the value of the call option
r is the risk-free interest rate
theta is the time decay of the option
S is the underlying stock price at time t
delta is a number between 0 and 1 which, in a hedging strategy, is the number of shares of a stock S held at time t.
sigma is the volatility of the stock, often written as (std dev price)*sqrt(T)
T is the expiration date
gamma = (N'(d1)e**(-q(T-t)))/(S(sigma)sqrt(T-1))
N' = (1/sqrt(2pi))e***1dt + sigma(t)dW
dr is the change in the short term interest rate
theta is a function describing how r moves which is designed to make such moves consistent with the zero-coupon yield curve of that day
alpha is the mean reversion rate (unsure what this is) 不明
sigma is the annual standard deviation of the short-term rate
dW is Brownian motion (a stochastic process which seems to have strong ties to the Gaussian distribution).

The Hull-White model is used to forecast interest rate changes and can thus be used to assess the value of interest rate derivatives (the security sense).
In the case of the Black-Scholes equation, I am not sure what derivatives(mathematical sense) are involved and what needs to be solved.
In the case of Hull-White, it seems that we want to integrate the right side with respect to r using two known boundary values of r, b1 and b2. Then we can equate the results of this definite integral with b2-b1 and solve for r. Actually, this would not work, as r is replaced with boundary values, so there is no r. Also, the r on the right is a known quantity (current rate). Thus, it probably makes the most sense to substitute for all parameters on the right and solve for dr, which is the move in the interest rate from today's known r.

*1:-x**2)/2) This equation tends to be applied to equity options, and the original model assumed stocks paid no dividends throughout the life of the option. The Hull-White model iswritten as: dr = (theta - alpha*r(t